Finite element formulation of the heat conduction equation in general orthogonal curvilinear coordinates
Abstract
A general approach to the finite element solution of heat conduction problems in orthogonal curvilinear coordinates is presented. The quasivariational approach is used to determine the functional which yields as an Euler equation the heat conduction equation in general orthogonal curvilinear coordinates and which accounts for boundary condition specification in a curvilinear frame. During the development, the time derivative of temperature is treated as a parameter when operating with the variational calculus. The resulting matrix differential equations can then be solved using finite differencing in the time domain. The developments of this work find important application to the class of problems for which the physical boundaries can be described by coordinate surfaces in an orthogonal curvilinear system. Two examples, employing the spherical and the oblate spheroidal coordinate systems, are included to illustrate the application of these results.
 Publication:

American Society of Mechanical Engineers
 Pub Date:
 November 1975
 Bibcode:
 1975asme.meetX....S
 Keywords:

 Calculus Of Variations;
 Conductive Heat Transfer;
 Finite Element Method;
 Partial Differential Equations;
 Convergence;
 Coordinates;
 Functionals;
 Orthogonal Functions;
 Time Dependence;
 Fluid Mechanics and Heat Transfer